Representation of the Convex Envelope of Bilinear Terms in a Reformulation Framework for Global Optimization

Abstract

Abstract Branch-and-bound in combination with convex underestimators constitute the basis for many algorithms in global nonconvex mixed-integer nonlinear programming (MINLP). Another option is to rely on reformulation-based techniques such as the α signomial global optimization (αSGO) algorithm, where power and exponential transformations for signomial or polynomial function and the α reformulation (αR) technique for general nonconvex twice-differentiable functions are used to reformulate the nonconvex problem. The transformations are approximated using piecewise linear functions (PLFs), resulting in a convex relaxation of the original nonconvex problem in an extended variable space. The solution to this reformulated problem provides a lower bound to the global minimum (of a minimization problem), and by iteratively refining the PLFs, the e-global solution can be obtained. A drawback with many reformulation-based techniques is that known convex envelopes cannot directly be utilized. However, in this paper, a formulation for expressing the convex envelope of bilinear terms in the αSGO framework is described, and it is shown that this improves the tightness of the lower bound.